We are given two sine curves with the following equations:

$y_1 = 20 \sin(2t - \frac{\pi}{6})$ $y_2 = 20 \sin(2t - \frac{7\pi}{6})$

(a) Find the phase of each curve

The general form of a sine function is:

$y = A \sin(Bt - \phi)$

Where:

• $A$ is the amplitude,
• $B$ is the angular frequency (related to the period),
• $\phi$ is the phase angle (or phase shift).

For the two curves, the phase angle $\phi$ is given by the term subtracted from $2t$.

1. Phase of $y_1$: The phase of $y_1 = 20 \sin(2t - \frac{\pi}{6})$ is $\frac{\pi}{6}$.
2. Phase of $y_2$: The phase of $y_2 = 20 \sin(2t - \frac{7\pi}{6})$ is $\frac{7\pi}{6}$.

(b) Find the phase difference between the curves

The phase difference $\Delta \phi$ between the two curves is the difference between the two phase angles:

$\Delta \phi = \left| \frac{7\pi}{6} - \frac{\pi}{6} \right|$

$\Delta \phi = \frac{6\pi}{6} = \pi$

(c) Determine whether the curves are in phase or out of phase

The curves are out of phase if the phase difference is not zero. In this case, the phase difference is $\pi$, which means the two sine waves are completely out of phase (they are 180 degrees apart). This means when one sine wave reaches its peak, the other is at its trough.

Summary of Results:

• (a) Phase of $y_1$: $\frac{\pi}{6}$
• (a) Phase of $y_2$: $\frac{7\pi}{6}$
• (b) Phase difference: $\pi$
• (c) The curves are out of phase.

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Do you want more details on phase shifts or have any questions? Here are five related questions to expand this topic:

1. How does the period of a sine wave relate to its angular frequency?
2. What happens to the phase shift if we increase the constant in the sine argument?
3. Can two sine waves with the same frequency ever be in phase if their phase difference is $\pi/2$?
4. What is the meaning of amplitude in the sine wave equations?
5. How does changing the amplitude affect the graph of a sine curve?

Tip: The phase difference of $\pi$ (180 degrees) between two sine waves means they will cancel each other out if added, leading to destructive interference.